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[资源] Quantum Mechanics in the Geometry of Space-Time

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Part I The Real Geometrical Algebra or Space–Time Algebra.
Comparison with the Language of the Complex Matrices
and Spinors
2 The Clifford Algebra Associated with the Minkowski
Space–Time M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 The Clifford Algebra Associated with an Euclidean Space . . . 7
2.2 The Clifford Algebras and the ‘‘Imaginary Number’’ ffiffiffiffiffiffiffi
p1 . . . . 9
2.3 The Field of the Hamilton Quaternions and the Ring
of the Biquaternion as Clþð3; 0Þ and Clð3; 0Þ ’ Clþð1; 3Þ. . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Comparison Between the Real and the Complex Language . . . . . 13
3.1 The Space–Time Algebra and the Wave Function
Associated with a Particle: The Hestenes Spinor . . . . . . . . . . 13
3.2 The Takabayasi–Hestenes Moving Frame . . . . . . . . . . . . . . . 15
3.3 Equivalences Between the Hestenes and the Dirac Spinors . . . 15
3.4 Comparison Between the Dirac and the Hestenes Spinors . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vii
Part II The U(1) Gauge in Complex and Real Languages.
Geometrical Properties and Relation with the Spin and
the Energy of a Particle of Spin 1/2
4 Geometrical Properties of the U(1) Gauge . . . . . . . . . . . . . . . . . . 21
4.1 The Definition of the Gauge and the Invariance of a Change
of Gauge in the U(1) Gauge . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.1 The U(1) Gauge in Complex Language. . . . . . . . . . . 21
4.1.2 The U(1) Gauge Invariance in Complex
Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.3 A Paradox of the U(1) Gauge in Complex
Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 The U(1) Gauge in Real Language . . . . . . . . . . . . . . . . . . . . 22
4.2.1 The Definition of the U(1) Gauge in Real
Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 The U(1) Gauge Invariance in Real Language . . . . . . 23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Relation Between the U(1) Gauge, the Spin and the Energy
of a Particle of Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 Relation Between the U(1) Gauge and the Bivector Spin . . . . 25
5.2 Relation Between the U(1) Gauge and the
Momentum–Energy Tensor Associated with the Particle . . . . . 25
5.3 Relation Between the U(1) Gauge and the Energy
of the Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Part III Geometrical Properties of the Dirac Theory
of the Electron
6 The Dirac Theory of the Electron in Real Language . . . . . . . . . . 29
6.1 The Hestenes Real form of the Dirac Equation . . . . . . . . . . . 29
6.2 The Probability Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.3 Conservation of the Probability Current. . . . . . . . . . . . . . . . . 30
6.4 The Proper (Bivector Spin) and the Total
Angular–Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.5 The Tetrode Energy–Momentum Tensor . . . . . . . . . . . . . . . . 31
6.6 Relation Between the Energy of the Electron and
the Infinitesimal Rotation of the ‘‘Spin Plane’’ . . . . . . . . . . . . 32
6.7 The Tetrode Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.8 The Lagrangian of the Dirac Electron . . . . . . . . . . . . . . . . . . 33
6.9 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
viii Contents
7 The Invariant Form of the Dirac Equation and Invariant
Properties of the Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.1 The Invariant Form of the Dirac Equation . . . . . . . . . . . . . . . 35
7.2 The Passage from the Equation of the Electron to the
One of the Positron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.3 The Free Dirac Electron, the Frequency and the Clock
of L. de Broglie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.4 The Dirac Electron, the Einstein Formula of the Photoeffect
and the L. de Broglie Frequency . . . . . . . . . . . . . . . . . . . . . 39
7.5 The Equation of the Lorentz Force Deduced from
the Dirac Theory of the Electron . . . . . . . . . . . . . . . . . . . . . 40
7.6 On the Passages of the Dirac Theory to the Classical Theory
of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Part IV The SU(2) Gauge and the Yang–Mills Theory in Complex
and Real Languages
8 Geometrical Properties of the SU(2) Gauge and the
Associated Momentum–Energy Tensor . . . . . . . . . . . . . . . . . . . . 45
8.1 The SU(2) Gauge in the General Yang–Mills Field Theory
in Complex Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 The SU(2) Gauge and the Y.M. Theory in STA. . . . . . . . . . . 46
8.2.1 The SU(2) Gauge and the Gauge Invariance
in STA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2.2 A Momentum–Energy Tensor Associated with
the Y.M. Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.2.3 The STA Form of the Y.M. Theory Lagrangian . . . . . 49
8.3 Conclusions About the SU(2) Gauge and the Y.M. Theory . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Part V The SU(2) 3 U(1) Gauge in Complex and Real Languages
9 Geometrical Properties of the SU(2) 3 U(1) Gauge . . . . . . . . . . . 53
9.1 Left and Right Parts of a Wave Function . . . . . . . . . . . . . . . 53
9.2 Left and Right Doublets Associated with
Two Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.3 The Part SU(2) of the SU(2) 9 U(1) Gauge. . . . . . . . . . . . . . 56
9.4 The Part U(1) of the SU(2) 9 U(1) Gauge . . . . . . . . . . . . . . 56
9.5 Geometrical Interpretation of the SU(2) 9 U(1) Gauge
of a Left or Right Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Contents ix
9.6 The Lagrangian in the SU(2) 9 U(1) Gauge. . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Part VI The Glashow–Salam–Weinberg Electroweak Theory
10 The Electroweak Theory in STA: Global Presentation. . . . . . . . . 61
10.1 General Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.2. The Particles and Their Wave Functions . . . . . . . . . . . . . . . . 62
10.2.1 The Right and Left Parts of the Wave Functions
of the Neutrino and the Electron . . . . . . . . . . . . . . . 62
10.2.2 A Left Doublet and Two Singlets . . . . . . . . . . . . . . . 62
10.3 The Currents Associated with the Wave Functions . . . . . . . . . 62
10.3.1 The Current Associated with the Right and Left
Parts of the Electron and Neutrino . . . . . . . . . . . . . . 63
10.3.2 The Currents Associated with the Left Doublet . . . . . 63
10.3.3 The Charge Currents . . . . . . . . . . . . . . . . . . . . . . . . 64
10.4 The Bosons and the Physical Constants. . . . . . . . . . . . . . . . . 65
10.4.1 The Physical Constants . . . . . . . . . . . . . . . . . . . . . . 65
10.4.2 The Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10.5 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
11 The Electroweak Theory in STA: Local Presentation. . . . . . . . . . 67
11.1 The Two Equivalent Decompositions of the Part LI
of the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
11.2 The Decomposition of the Part LII of the Lagrangian into
a Charged and a Neutral Contribution . . . . . . . . . . . . . . . . . . 68
11.2.1 The Charged Contribution . . . . . . . . . . . . . . . . . . . . 69
11.2.2 The Neutral Contribution. . . . . . . . . . . . . . . . . . . . . 69
11.3 The Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
11.3.1 The Part U(1) of the SU(2) 9 U(1) Gauge . . . . . . . . 70
11.3.2 The Part SU(2) of the SU(2) 9 U(1) Gauge . . . . . . . 71
11.3.3 Zitterbewegung and Electroweak Currents
in Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Part VII On a Change of SU(3) into Three SU(2) 3 U(1)
12 On a Change of SU(3) into Three SU(2) 3 U(1) . . . . . . . . . . . . . 75
12.1 The Lie Group SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
12.1.1 The Gell–Mann Matrices ka. . . . . . . . . . . . . . . . . . . 75
x Contents
12.1.2 The Column W on which the Gell–Mann
Matrices Act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.1.3 Eight Vectors Ga . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.1.4 A Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.1.5 On the Algebraic Nature of the Wk . . . . . . . . . . . . . . 76
12.1.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
12.2 A passage From SU(3) to Three SU(2) 9 U(1) . . . . . . . . . . . 77
12.3 An Alternative to the Use of SU(3) in Quantum
Chromodynamics Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Part VIII Addendum
13 A Real Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 83
13.1 General Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
13.2 Electromagnetism: The Electromagnetic Potential. . . . . . . . . . 84
13.2.1 Principles on the Potential . . . . . . . . . . . . . . . . . . . . 84
13.2.2 The Potential Created by a Population of Charges . . . 85
13.2.3 Notion of Charge Current . . . . . . . . . . . . . . . . . . . . 86
13.2.4 The Lorentz Formula of the Retarded Potentials. . . . . 87
13.2.5 On the Invariances in the Formula of the
Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 88
13.3 Electrodynamics: The Electromagnetic Field,
the Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
13.3.1 General Definition . . . . . . . . . . . . . . . . . . . . . . . . . 89
13.3.2 Case of Two Punctual Charges: The Coulomb Law . . 89
13.3.3 Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . 90
13.3.4 Electric and Magnetic Fields Deduced from the
Lorentz Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 91
13.3.5 The Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . 93
13.4 Electrodynamics in the Dirac Theory of the Electron . . . . . . . 93
13.4.1 The Dirac Probability Currents. . . . . . . . . . . . . . . . . 94
13.4.2 Current Associated with a Level E of Energy . . . . . . 94
13.4.3 Emission of an Electromagnetic Field . . . . . . . . . . . . 95
13.4.4 Spontaneous Emission. . . . . . . . . . . . . . . . . . . . . . . 95
13.4.5 Interaction with a Plane Wave . . . . . . . . . . . . . . . . . 96
13.4.6 The Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Contents xi
Part IX Appendices
14 Real Algebras Associated with an Euclidean Space . . . . . . . . . . . 105
14.1 The Grassmann (or Exterior) Algebra of Rn . . . . . . . . . . . . . 105
14.2 The Inner Products of an Euclidean Space E ¼ Rq;nq . . . . . . 105
14.3 The Clifford Algebra CIðEÞ Associated with
an Euclidean Space E ¼ Rp;np . . . . . . . . . . . . . . . . . . . . . . 106
14.4 A Construction of the Clifford Algebra . . . . . . . . . . . . . . . . . 108
14.5 The Group OðEÞ in CIðEÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 109
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
15 Relation Between the Dirac Spinor and the Hestenes Spinor . . . . 111
15.1 The Pauli Spinor and Matrices . . . . . . . . . . . . . . . . . . . . . . . 111
15.2 The Dirac spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
15.3 The Quaternion as a Real Form of the Pauli spinor . . . . . . . . 113
15.4 The Biquaternion as a Real Form of the Dirac spinor . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
16 The Movement in Space–Time of a Local
Orthonormal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.1 C.1 The Group SOþðEÞ and the Infinitesimal
Rotations in ClðEÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.2 Study on Properties of Local Moving Frames . . . . . . . . . . . . 116
16.3 Infinitesimal Rotation of a Local Frame . . . . . . . . . . . . . . . . 116
16.4 Infinitesimal Rotation of Local Sub-Frames . . . . . . . . . . . . . . 117
16.5 Effect of a Local Finite Rotation of a Local Sub-Frame . . . . . 118
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
17 Incompatibilities in the Use of the Isospin Matrices . . . . . . . . . . . 121
17.1 W is an ‘‘Ordinary’’ Dirac Spinor . . . . . . . . . . . . . . . . . . . . . 121
17.2 W is a Couple (Wa; Wb) of Dirac Spinors . . . . . . . . . . . . . . . 121
17.3 W is a Right or a Left Doublet. . . . . . . . . . . . . . . . . . . . . . . 122
17.4 Questions about the Nature of the Wave Function . . . . . . . . . 122
18 A Proof of the Tetrode Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 123
19 About the Quantum Fields Theory . . . . . . . . . . . . . . . . . . . . . . . 125
19.1 On the Construction of the QFT. . . . . . . . . . . . . . . . . . . . . . 125
19.2 Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
19.3 An Artifice in the Lamb Shift Calculation . . . . . . . . . . . . . . . 127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

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